Review and perspective on mathematical modelingof microbial ecosystemsAntonella Succurro1,2 and Oliver Ebenhöh2,3
1Botanical Institute, University of Cologne, Cologne, Germany; 2Cluster of Excellence on Plant Sciences (CEPLAS), Düsseldorf, Germany; 3Institute of Quantitative and TheoreticalBiology, Heinrich-Heine-University, Düsseldorf, Germany
Understanding microbial ecosystems means unlocking the path toward a deeper knowl-edge of the fundamental mechanisms of life. Engineered microbial communities are alsoextremely relevant to tackling some of today’s grand societal challenges. Advancedmeta-omics experimental techniques provide crucial insights into microbial communities,but have been so far mostly used for descriptive, exploratory approaches to answer theinitial ‘who is there?’ question. An ecosystem is a complex network of dynamic spatio-temporal interactions among organisms as well as between organisms and the environ-ment. Mathematical models with their abstraction capability are essential to capture theunderlying phenomena and connect the different scales at which these systems act.Differential equation models and constraint-based stoichiometric models are deterministicapproaches that can successfully provide a macroscopic description of the outcomefrom microscopic behaviors. In this mini-review, we present classical and recent applica-tions of these modeling methods and illustrate the potential of their integration. Indeed,approaches that can capture multiple scales are needed in order to understand emergentpatterns in ecosystems and their dynamics regulated by different spatio-temporalphenomena. We finally discuss promising examples of methods proposing the integrationof differential equations with constraint-based stoichiometric models and argue that morework is needed in this direction.
IntroductionThe relevance of bacteria and microbes for life as we know it cannot be exaggerated: they were thefirst forms of life that drastically shaped Earth’s environment, and animal life evolved in a microbe-dominated landscape. After being considered for decades mostly as pathogens, they are now receivingfull credit for the fundamental positive role they often play in ecosystems [1]. Many different bio-logical complexes can fall under the term ‘ecosystem’, and similar techniques can be employed tostudy a microbial mat in the Yellowstone National Park and the gut microbiome of the authors of thismanuscript. In general terms, an ecosystem consists of an ensemble of organisms, their shared envir-onment and the complex network of resulting interactions, either among the organisms or betweenorganisms and environment. Depending on the particular interest of the observer, different questionscan be addressed with different empirical and theoretical methods.Meta-omics technologies have been so far mostly used for descriptive, exploratory approaches to
answer the initial ‘who is there?’ question. Obtaining this sort of ‘stamp collection’ [2] is a necessarybut not sufficient condition to build a comprehensive model of how microbial communities assemble,maintain themselves, evolve and function. Once we realize that macroscopic features are the emergingobservables of microscopic interactions [3], the need to obtain a mechanistic understanding of theecosystem dynamics becomes evident. At the same time, it is clear that the same ecosystem is regu-lated by phenomena acting at very different scales, both in space and time (Figure 1). Observables likegrowth rates might be typically associated to one specific scale (say, population-level ecology), but are
Version of Record published:14 March 2018
Received: 18 September 2017Revised: 24 November 2017Accepted: 5 February 2018
not independent from others (say, individual metabolic states). Mathematical models with their abstraction cap-ability become an essential tool to capture specific phenomena and connect different scales [4].This mini-review focuses on two major classes of mathematical methods (differential equation models and
constraint-based stoichiometric models) and their recent applications to the study of microbial communities.After highlighting the individual strengths and weaknesses of these methods, attention is given to integrativeapproaches proposed or desirable in view of obtaining a more comprehensive representation of biologicalsystems. For a broader overview of mathematical modeling of microbial communities, the reader is referred to[5,6]. As a general note, it is worth to point out that this mini-review is describing two deterministic modelingtechniques. For the sake of brevity and of uniformity of the manuscript, stochastic models are not consideredhere, but the reader should be aware that stochasticity is an intrinsic property of Nature. However, as previouslymentioned, depending on the biological system under study and the question addressed, different methodscan be chosen. Deterministic models can be considered as the macroscopic description of the outcome frommicroscopic behaviors.
Ecology and differential equation modelsThe study of emergent patterns in ecosystems and their dynamics on multiple spatio-temporal scales is acentral focus in the well-established field of theoretical ecology [7]. Microbial ecological systems biology allowsinvestigating in silico, in vivo and in vitro most of the temporal scales, including evolutionary dynamics. Intheir recent review, Friedman and Gore [8] highlight how models implementing simple qualitative interactions(competition, cooperation and exploitation) can still have high predictive power.Almost two centuries ago, Verhulst defined a single deterministic equation to describe population growth
which is also capturing population-level behavior of bacterial cultures [9]. Since Lotka [10] and Volterra [11]independently proposed a mathematical model of population dynamics based on ordinary differential equations(ODEs), Lotka–Volterra (LV) models have been widely used to describe the time evolution of ecosystems(Figure 2). In the classical predator–prey LV model, two competing populations directly affect each other’sgrowth, either positively or negatively. The model parameters, including the initial conditions, quantify suchdirect interactions and will determine the stability of the system. Indeed, these mathematical systems of equa-tions can be numerically solved and the qualitative properties of the steady-state solutions, from periodic oscil-lations to chaotic attractors, can be evaluated.In 1987, Hofbauer et al. [12] extended the LV equations to an arbitrary number of coexisting populations
and studied the mathematical properties of these generalized LV (gLV) models. Figure 2 shows schematicallythat today a gLV model can be obtained from a time series of metagenomics survey used to infer aco-occurrence network in a rather straightforward way [13]. Owing to the technological advances in sequencingtechniques, gLV models have been successfully used over the past few years to study the temporal dynamics ofvarious bacterial communities [14,15]. Their predictive power is, however, still to be demonstrated, especially
Figure 1. Examples of different levels of complexity acting at different scales.
Ecosystems span very different levels of complexity and of temporal and spatial scales. The biological question sets the
importance of each aspect and defines the abstraction needed for a mathematical representation. Experimental observables
also strongly influence the design of input and output of a theoretical model.
considering the reductive assumption that the community dynamics is driven by only pairwise interactions andthat the environmental conditions are not taken into account.A promising step forward in gLV model development is the approach by Stein et al. [16]. In a study on col-
onization of mice gut microbiome by the pathogen Clostridium difficile, they added to the standard gLV formu-lation (individual growth rates and an interaction matrix) a third term that models individual susceptibility toan environmental perturbation, antibiotic administering, in their case study (Figure 2). Time series of in vivometagenomics data followed the community composition of the intestinal microbiome of mice under threeconditions: unperturbed and infected with C. difficile; perturbed by antibiotic; perturbed and then infected.After discretizing the gLV equations, they could use regularized linear regression to fit the parameters over atraining set of data. The obtained gLV model was then used to predict the community behavior in the condi-tions left out of the training set. Steady-state analysis correctly predicted the infection outcomes in terms ofcommunity profiles and composition and allowed for exploration of alternative stable configurations.To capture spatial effects like diffusion of nutrients and cell motility [17], partial differential equations
(PDEs) are commonly applied. In wastewater treatment, the activated sludge system is a process where themicrobial biomass is employed to perform specific biological tasks, like removal of N and P from sewage. Inthis context, models based on differential equations have been developed since the 1980s aimed at aiding thedesign of industrial plants [18]. An early example of an ODE and PDE model is from Benefield and Molz [19],which described the ecosystem composed by an aggregated microbial suspension (floc particles) and solublemetabolites with a system of five PDEs and four ODEs. Under the assumption that organic C, O, N and P aregrowth-limiting nutrients and that their transfer within a floc particle is controlled by molecular diffusion, theyprovide a model that can be tested under different operating conditions, such as oxygenation levels, and whichallows assessing, e.g. the influence of including a nitrifying microbe on nutrient limitation.In conclusion of this section, building on time series of metagenomics data, gLV models can potentially
address the question ‘who will be there?’, but their power is generally limited to a data-driven inference of pair-wise organism interactions in a certain environment. The reality is far more complex and it is hard to imagine
Figure 2. Modeling microbial communities with ODE systems.
Microbial growth over time follows the logistic rule defined by Verhulst in 1838. Predator–prey systems show an oscillatory
dynamic. gLV models are used today in combination with time series of metagenomics data. Stein et al. [16] proposed an
extension of the gLV model to include susceptibility to an external time-dependent perturbation.
that we will ever answer questions like ‘who does what?’, or ‘what will happen if…?’, without sound mathemat-ical models offering mechanistic insights into the various aspects regulating and characterizing the metabolicstate of a microbial community. ODE and PDE models can successfully capture biochemical activity andspatial heterogeneity, but rely on a priori assumptions on the mechanisms and on known parameters.Metabolism is well known to be a key driver of community interactions [20], and differential equation modelsof large metabolic pathways have been proposed [21]. While technically not impossible, a genome-scale differ-ential equation model of an organism’s full metabolism is not practical. Such a model would consist of asystem of thousands of equations, each of which requiring empirical knowledge, or inference, of several para-meters. This approach cannot keep up with the speed at which new genomes are published and will not helptoward a thorough exploration of emerging metabolic properties of ecosystems. As of today, the most conveni-ent method to model genome-scale metabolism is through metabolic network reconstruction and constraint-based stoichiometric models.
Metabolism and constraint-based modelsGenome sequencing technology has opened the door for understanding the building blocks of life. There isstill, however, a significant gap between what we are able to observe in the genome and what genes can berelated to known functions. Our current knowledge of enzymatic activity associated to amino acid sequences isfar from complete, but also at the same time is constantly expanding and collected into searchable databases,making it easily accessible to automated computational pipelines. The recent years have seen large improve-ments in the process of reconstructing operational genome-scale metabolic models [22], also owing to plat-forms like KBase [23]. An example workflow for the generation and analysis of a genome-scale metabolicmodel is described in Figure 3. It is important to point out that consensus on a unified standard is highlyneeded to ensure the reproducibility and reusability of published genome-scale networks [24].Once a network of biochemical reactions is known, it can be mathematically represented as a stoichiometric
matrix S of dimension (m, r), where m is the number of metabolites and r is the number of reactions in themodel, and the elements of the matrix sij are the stoichiometric coefficients of metabolite i in reaction j [25].The mass balance of intracellular metabolites translates into the set of differential equations resulting from theproduct of S and the vector of reaction fluxes v (Figure 3). Constrained-based models (CBMs) assume a steadystate and impose constraints on reaction flux values based on thermodynamic considerations (e.g. irreversibilityof some reactions) and eventually biological or experimental information [26,27]. Since the dimension m is ingeneral lower than the dimension r, the resulting space of solutions for the vector of reaction fluxes v is aconvex polyhedral cone, also called flux cone. The metabolic flux distribution reflects a certain cellular state,possibly observable as a phenotype under the defined boundary conditions. The constraints on the reactionflux of import reactions (i.e. reactions transporting metabolites in the system, r1 and r2 in Figure 3) typicallyrepresent nutrient availability.Different methods have been developed to study the structure and functionality of genome-scale metabolic
networks (Figure 3). Elementary mode analysis (EMA) solves the CBM equations to identify the set of all pos-sible unique and minimal pathways that allow steady-state metabolic fluxes in the network [28,29]. EMA per-forms a computationally expensive calculation, especially on large genome-scale metabolic networks, but bringsrigorous insights into the structure of a network. Pathways can then be accurately and systematically comparedand their efficiency, e.g. in terms of molar yields of a product, can be easily assessed. It is important to high-light that EMA does not rely on a priori assumptions on the network except for thermodynamics constraints.Since the computed solutions are scalable, environmental conditions like nutrient availability can be subse-quently used, e.g. to define an intake flux and normalize the overall flux distribution. A cellular physiologicalstate is then a weighted linear combination of elementary modes.Flux balance analysis (FBA) is a convenient method to obtain a single flux distribution in fast computation
time [30,31]. FBA assumes the previously described CBM constraints and adds the declaration of an objectivefunction which, if linear in the metabolic fluxes, sets the definition of an optimization problem solvable withlinear programming (LP). Typical objective functions are, e.g. maximization of growth rate, maximization orminimization of ATP production, minimization of overall fluxes. As a consequence, however, the resulting fluxdistribution depends on the modeler’s subjective choice of the objective function [32]. Furthermore, alternativeoptima can exist for the same problem, but FBA will return a unique solution. A plethora of extensions to FBAhave been proposed in the past decades and ongoing efforts are particularly focused on integrating informationfrom large meta-omics datasets to further constrain the models with biological data. FBA and its variants have
been successfully applied to predict phenotypes of biological systems, e.g. resulting from gene knock-outs andproduct yield optimization, but most of them rely on strict assumptions and often trade the computationallightness typical of FBA for more accurate solutions [33]. A critical review of the underlying assumptions inFBA and related methods is presented, e.g. in [34].The so far described approaches have been initially applied, obviously, to single organisms. Scientists are still
working on understanding the secrets encoded in an individual genome, but nevertheless it is possible to usemetabolic network modeling also to explore the properties of natural and synthetic microbial communities.The first step to take is the definition of the community-level metabolic network model itself, and Henry et al.[35] recently proposed an effective strategy for a data-driven network reconstruction. In Table 1, we showthree commonly chosen configurations: lumped or supra-organism network; compartmentalized network; inde-pendent multispecies networks. These approaches build on different assumptions and have specific advantages
Figure 3. Example workflow for genome-scale metabolic network reconstruction and analysis with FBA.
The process of reconstructing and analyzing a genome-scale metabolic network model starts with a sequenced genome.
Functional annotation of the genome [50] links genes to enzymatic activity and allows the reconstruction of a draft network of
metabolic reactions. Further steps include compartmentalization, the addition of exchange and transport reactions, the
definition of a biomass equation and the gap filling procedure. Gap filling is needed to complement pathways where enzymes
are missing, usually because of incomplete annotation knowledge [22]. Today automated workflows like the Model SEED [51]
and KBase [23] allow quick reconstruction of genome-scale metabolic network models, but do not solve yet the eventual need
for manual curation. The network of reactions can then be represented mathematically as a stoichiometric matrix and analyzed
with CBMs under the steady-state assumption and imposing boundaries on the reaction fluxes. Elementary modes [29] and
FBA [31] are widely used methods to study the metabolic flux distributions.
and disadvantages. It is hence up to the modelers to choose the method most suitable for the biologicalproblem at hand. For example, Khandelwal et al. [36] proposed an extension of FBA that with the singleassumption of balanced growth (biologically justified in controlled environments where cultures are stabilizedat logarithmic growth phases) could predict metabolic fluxes, community growth rate and individual biomassabundances of a compartmentalized community metabolic network model. Other approaches introduce add-itional assumptions, like a community-level objective or specific functional roles for each community member,thus reducing the range of biological systems that can be modeled to those that can be characterized at therequired level. This, however, does not imply that such models are not useful; only that there is a need for newapproaches based on ideally universal principles. Since, in general, CBMs of communities have been extensivelyreviewed (see, e.g. [37–39]), we only highlight some properties and examples of the possible model configura-tions in Table 1.Recently, Beck et al. [40] studied the ecological acclimation to stresses from high irradiance, O2/CO2 compe-
tition and nutrient limitation in the thermophilic cyanobacterium Thermosynechococcus elongatus BP-1 withEMA and resource allocation theory. T. elongatus is often found in nature in association with heterotrophs inbacterial mats and the authors could assess the impact of stress acclimation on the cyanobacterium’s commu-nity interactions. Indeed, the analysis of the phenotypic space revealed that reduced carbon byproducts aresecreted under environmental stress, creating a favorable ecological niche for heterotrophs. Cross-feeding is apossible strategy for stress relief, as aerobic heterotrophs like Meiothermus ruber would consume O2, thus low-ering O2/CO2 competition, and organic acids, thus preventing inhibitory effects. The predicted heterotroph–photoautotroph ratio as a function of stress acclimation was found to be in accordance with measured data.The previous example showed how EMA on a single organism can reveal ecosystem configurations favorable
for natural community establishment. Another approach to study microbial consortia is to assemble a multi-organism stoichiometric model (see Table 1) and investigate the interdependencies of the community memberswith FBA. Koch et al. [41] used the compartmentalized method to model a bacterial community of threespecies of industrial interest for biogas production: Desulfovibrio vulgaris, Methanococcus maripaludis andMethanosarcina barkeri. After assessing the performance of single organism metabolic network models, theauthors built a community model with a hierarchical objective for FBA: maximal community growth rate andmaximal individual biomass yield. By quantifying the degree of optimality as the ratio between the communitygrowth rate and its expected value when all species use substrates optimally for biomass production, they couldinvestigate different community behaviors. Simulations under different substrate utilizations predicted optimalcommunity compositions and related it to methane yield. This information, to be validated experimentally, isan example of how CBMs can be of interest for industrial microbial community design.
Table 1 Examples of community metabolic network analysis strategiesA community-level metabolic network model can be defined in different ways. Three main approaches are shown here: lumped,compartmentalized and independent networks.
Besides the limitations caused by necessary biological assumptions mentioned earlier, in the context ofmathematical modeling of ecosystems the main disadvantage of CBMs is the loss, by construction, ofdynamic information. To a certain extent, re-integration of dynamic interaction with external (environmental)nutrient availability is straightforward in the dynamic FBA (dFBA) approach [42]. Exchange reactions areincluded in the metabolic network and their flux values as determined by FBA represent either import orexport of a metabolite. By discretizing time in intervals and assuming quasi-steady-state conditions (i.e. meta-bolism adjusts quickly to external perturbations), it is possible to update at each time step the metabolitelevels based on how much has been consumed or secreted. In the next iteration, the flux boundaries forthe subsequent FBA problem are accordingly updated and the process continues. In the next section, wediscuss approaches expanding this concept and the need for integration of dynamic and structural models tomodel ecosystems.
Integration approachesFigure 4 shows a schematic representation of how CBMs and dynamic equations can be integrated. Over thelast years, promising approaches to extend dFBA to multi-organism and environmental temporal dynamicshave been proposed. The early fundamental works have been previously reviewed, e.g. in [37–39]. We willtherefore only briefly highlight some important publications in which a considerable methodological advancewas presented. First, Zhuang et al. [43] developed a Dynamic Multi-species Metabolic Modeling framework tocouple the CBMs of a Geobacter and a Rhodoferax with a groundwater environment and simulate bacterialcompetition; then Zomorrodi et al. [44] proposed a multi-objective framework, d-OptCom, that introducedindividual- and community-level fitness principles as inner and outer optimization problems, respectively;going a step further, Harcombe et al. [45] added spatial dynamics to the temporal evolution of communities bysimulating dFBA on a lattice. More recently, Phalak et al. [46] obtained a temporal and spatial representation
Figure 4. Example integration of CBMs and dynamic equations.
Methods like FBA provide a metabolic flux distribution at steady state. Assuming a quasi-steady state, it is possible to interface
FBA with ODEs to capture temporal environmental changes (typically, nutrient availability) and growth dynamics. The spatial
component, in particular in terms of particle diffusion, can be obtained by integrating FBA with PDEs. Space can be discretized
to reduce the computational cost of the simulation.
with genome-scale resolution of a community of bacteria found in chronic wound biofilm, Pseudomonas aeru-ginosa and Staphylococcus aureus. The FBA problem is formulated as a series of LP problems with differentobjectives (maximizing growth, minimizing byproduct secretion and maximizing consumption of key nutrients)to match biological observations or assumptions. By implementing PDEs to describe the convective and diffu-sional processes in the biofilm layer, the authors could predict the spatial partitioning of the two species. Themodel also allowed assessing the impact of nutrient competition, cross-feeding and inhibition of S. aureus by asmall molecule secreted by P. aeruginosa.All the examples cited so far, relying on more or less stringent biological assumptions, are able to capture
certain aspects of ecosystem dynamics directly driven by metabolic interactions like cross-feeding or competi-tion. Assuming that automated reconstruction of functional genome-scale CBMs will substantially improve inthe near future, major critical points are still to be addressed. Relevant mechanisms of community behavior arelost because of intrinsic limitations of CBMs (e.g. cofactor dependence of enzymatic activities is modeled byhardcoding cofactors in the biomass function). The implementation of bi-level objective functions still forcesthe microbes to act following subjective assumptions that might be specific for only certain experimental condi-tions. The phenomena modeled are in general relative to only organism-level scales, and little abstraction isused to simplify the problem under study, which is one of the strengths of mathematical models.
PerspectivesThe technical capability to sequence natural microbial communities including strains that are not culturable inthe laboratory opens up a fascinating scenario of possible novel discoveries. Furthermore, a deeper understand-ing of the mechanisms that regulate and stabilize ecosystems is becoming crucial today to address grand societalchallenges like food security and bioremediation.Theoretical ecology has developed sound mathematical methods to understand emergent dynamics of
ecosystems. Dynamic models have the advantage of being easily applied to different spatio-temporal scales, butthey often require educated guesses on the fundamental processes to be examined. They are therefore notsuitable for systematic surveys of the metabolic capabilities of microbial communities. Multi-organismgenome-scale metabolic modeling is today still challenging and heavily based on a priori assumptions, but it isalso a rapidly developing field. While still in its infancy, it already now offers insights into the structuralproperties of metabolism and allows to study pathway optimization and strain engineering, and it is becominga major investigation tool, owing also to the significant efforts from the systems biology community to integrateit with experimental data. However, genome-scale metabolic modeling alone ignores a fundamental aspect ofecosystems: the cellular response to dynamic environmental changes. These include processes with similar localeffects but acting at diverse spatio-temporal scales [4], like changes in nutrient availability caused by global geo-chemical cycles (e.g. the nitrogen cycle) or man-made perturbations (e.g. artificial fertilizers).Recently, there has been rising interest in modeling approaches that integrate different methods, and in this
manuscript we focused on two of them. We believe that current theoretical tools can achieve a much higherpredictive power by following two principles: simplicity and scalability. By recognizing key strengths of specificmethods and integrating them to represent multi-scale phenomena, it will be possible to disentangle thecomplex web of interactions in microbial ecosystem and engineer synthetic communities.
References1 Widder, S., Allen, R.J., Pfeiffer, T., Curtis, T.P., Wiuf, C., Sloan, W.T. et al. (2016) Challenges in microbial ecology: building predictive understanding of
community function and dynamics. ISME J. 10, 2557–2568 https://doi.org/10.1038/ismej.2016.452 Yong, E. (2016) I Contain Multitudes: The Microbes Within us and a Grander View of Life, Ecco, New York, NY3 Schrödinger, E. (1944) What is life? The physical aspect of the living cell, Cambridge University Press, Cambridge, UK4 Succurro, A., Moejes, F.W. and Ebenhöh, O. (2017) A diverse community to study communities: integration of experiments and mathematical models to
study microbial consortia. J. Bacteriol. 199, e00865-16 https://doi.org/10.1128/JB.00865-165 Song, H.S., Cannon, W., Beliaev, A. and Konopka, A. (2014) Mathematical modeling of microbial community dynamics: a methodological review.
Processes 2, 711–752 https://doi.org/10.3390/pr20407116 Zomorrodi, A.R. and Segrè, D. (2016) Synthetic ecology of microbes: mathematical models and applications. J. Mol. Biol. 428, 837–861 https://doi.org/
10.1016/j.jmb.2015.10.0197 Hagstrom, G.I. and Levin, S.A. (2017) Marine ecosystems as complex adaptive systems: emergent patterns, critical transitions, and public goods.
Ecosystems 20, 458–476 https://doi.org/10.1007/s10021-017-0114-38 Friedman, J. and Gore, J. (2017) Ecological systems biology: the dynamics of interacting populations. Curr. Opin. Syst. Biol. 1, 114–121 https://doi.org/
10.1016/j.coisb.2016.12.0019 Verhulst, P.H. (1838) Notice sur la loi que la population poursuit dans son accroissement. Corresp. Math. Phys. 10, 113–12110 Lotka, A.J. (1925) Elements of Physical Biology. Williams and Wilkins11 Volterra, V. (1926) Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 https://doi.org/10.1038/118558a012 Hofbauer, J., Hutson, V. and Jansen, W. (1987) Coexistence for systems governed by difference equations of Lotka–Volterra type. J. Math. Biol. 25,
553–570 https://doi.org/10.1007/BF0027619913 Berry, D. and Widder, S. (2014) Deciphering microbial interactions and detecting keystone species with co-occurrence networks. Front. Microbiol. 5,
219 https://doi.org/10.3389/fmicb.2014.0021914 Mounier, J., Monnet, C., Vallaeys, T., Arditi, R., Sarthou, A.S., Hélias, A. et al. (2008) Microbial interactions within a cheese microbial community. Appl.
Environ. Microbiol. 74, 172–181 https://doi.org/10.1128/AEM.01338-0715 Faust, K. and Raes, J. (2012) Microbial interactions: from networks to models. Nat. Rev. Microbiol. 10, 538–550 https://doi.org/10.1038/nrmicro283216 Stein, R.R., Bucci, V., Toussaint, N.C., Buffie, C.G., Rätsch, G., Pamer, E.G. et al. (2013) Ecological modeling from time-series inference: insight into
dynamics and stability of intestinal microbiota. PLoS Comput. Biol. 9, e1003388 https://doi.org/10.1371/journal.pcbi.100338817 Tindall, M.J., Maini, P.K., Porter, S.L. and Armitage, J.P. (2008) Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial
populations. Bull. Math. Biol. 70, 1570–1607 https://doi.org/10.1007/s11538-008-9322-518 Hauduc, H., Rieger, L., Oehmen, A., van Loosdrecht, M.C.M., Comeau, Y., Héduit, A. et al. (2013) Critical review of activated sludge modeling: state of
process knowledge, modeling concepts, and limitations. Biotechnol. Bioeng. 110, 24–46 https://doi.org/10.1002/bit.2462419 Benefield, L. and Molz, F. (1983) A kinetic model for the activated sludge process which considers diffusion and reaction in the microbial floc.
Biotechnol. Bioeng. 25, 2591–2615 https://doi.org/10.1002/bit.26025110920 Zelezniak, A., Andrejev, S., Ponomarova, O., Mende, D.R., Bork, P. and Patil, K.R. (2015) Metabolic dependencies drive species co-occurrence in
diverse microbial communities. Proc. Natl Acad. Sci. U.S.A. 112, 6449–6454 https://doi.org/10.1073/pnas.142183411221 Smallbone, K., Simeonidis, E., Swainston, N. and Mendes, P. (2010) Towards a genome-scale kinetic model of cellular metabolism. BMC Syst. Biol. 4,
6 https://doi.org/10.1186/1752-0509-4-622 Fell, D.A., Poolman, M.G. and Gevorgyan, A. (2010) Building and analysing genome-scale metabolic models. Biochem. Soc. Trans. 38, 1197–1201
https://doi.org/10.1042/BST038119723 Arkin, A.P., Stevens, R.L., Cottingham, R.W., Maslov, S., Henry, C.S., Dehal, P. et al. (2016) The DOE Systems Biology Knowledgebase (KBase). bioRxiv
https://doi.org/10.1101/09635424 Ravikrishnan, A. and Raman, K. (2015) Critical assessment of genome-scale metabolic networks: the need for a unified standard. Brief. Bioinform. 16,
1057–1068 https://doi.org/10.1093/bib/bbv00325 Heinrich, R. and Schuster, S. (1996) The Regulation of Cellular Systems, Springer, Boston, MA26 Fell, D.A. and Small, J.R. (1986) Fat synthesis in adipose tissue. An examination of stoichiometric constraints. Biochem. J. 238, 781–786 https://doi.
org/10.1042/bj238078127 Varma, A. and Palsson, B.O. (1993) Metabolic capabilities of Escherichia coli II. Optimal growth patterns. J. Theor. Biol. 165, 503–522 https://doi.org/
10.1006/jtbi.1993.120328 Schuster, S. and Hilgetag, C. (1994) On elementary flux modes in biochemical reaction systems at steady state. J. Biol. Syst. 02, 165–182 https://doi.
org/10.1142/S021833909400013129 Trinh, C.T., Wlaschin, A. and Srienc, F. (2009) Elementary mode analysis: a useful metabolic pathway analysis tool for characterizing cellular
metabolism. Appl. Microbiol. Biotechnol. 81, 813–826 https://doi.org/10.1007/s00253-008-1770-130 Varma, A. and Palsson, B.O. (1994) Stoichiometric flux balance models quantitatively predict growth and metabolic by-product secretion in wild-type
Escherichia coli W3110. Appl. Environ. Microbiol. 60, 3724–3731 PMID:798604531 Orth, J.D., Thiele, I. and Palsson, B.Ø. (2010) What is flux balance analysis? Nat. Biotechnol. 28, 245–248 https://doi.org/10.1038/nbt.161432 Schuetz, R., Kuepfer, L. and Sauer, U. (2007) Systematic evaluation of objective functions for predicting intracellular fluxes in Escherichia coli. Mol. Syst.
Biol. 3, 119 https://doi.org/10.1038/msb410016233 Lewis, N.E., Nagarajan, H. and Palsson, B.O. (2012) Constraining the metabolic genotype–phenotype relationship using a phylogeny of in silico methods.
Nat. Rev. Microbiol. 10, 291–305 https://doi.org/10.1038/nrmicro273734 Schuster, S., Pfeiffer, T. and Fell, D.A. (2008) Is maximization of molar yield in metabolic networks favoured by evolution? J. Theor. Biol. 252, 497–504
https://doi.org/10.1016/j.jtbi.2007.12.00835 Henry, C.S., Bernstein, H.C., Weisenhorn, P., Taylor, R.C., Lee, J.Y., Zucker, J. et al. (2016) Microbial community metabolic modeling: a community
data-driven network reconstruction. J. Cell. Physiol. 231, 2339–2345 https://doi.org/10.1002/jcp.25428
36 Khandelwal, R.A., Olivier, B.G., Röling, W.F.M., Teusink, B. and Bruggeman, F.J. (2013) Community flux balance analysis for microbial consortia atbalanced growth. PLoS ONE 8, e64567 https://doi.org/10.1371/journal.pone.0064567
37 Perez-Garcia, O., Lear, G. and Singhal, N. (2016) Metabolic network modeling of microbial interactions in natural and engineered environmental systems.Front. Microbiol. 7, 673 https://doi.org/10.3389/fmicb.2016.00673
38 Bosi, E., Bacci, G., Mengoni, A. and Fondi, M. (2017) Perspectives and challenges in microbial communities metabolic modeling. Front. Genet. 8, 88https://doi.org/10.3389/fgene.2017.00088
39 Hanemaaijer, M., Roeling, W.F.M., Olivier, B.G., Khandelwal, R.A., Teusink, B. and Bruggeman, F.J. (2015) Systems modeling approaches for microbialcommunity studies: from metagenomics to inference of the community structure. Front. Microbiol. 6, 213 https://doi.org/10.3389/fmicb.2015.00213
40 Beck, A., Bernstein, H. and Carlson, R. (2017) Stoichiometric network analysis of cyanobacterial acclimation to photosynthesis-associated stressesidentifies heterotrophic niches. Processes 5, 32 https://doi.org/10.3390/pr5020032
41 Koch, S., Benndorf, D., Fronk, K., Reichl, U. and Klamt, S. (2016) Predicting compositions of microbial communities from stoichiometric models withapplications for the biogas process. Biotechnol. Biofuels 9, 17 https://doi.org/10.1186/s13068-016-0429-x
42 Mahadevan, R., Edwards, J.S. and Doyle, F.J. (2002) Dynamic flux balance analysis of diauxic growth in Escherichia coli. Biophys. J. 83, 1331–1340https://doi.org/10.1016/S0006-3495(02)73903-9
43 Zhuang, K., Izallalen, M., Mouser, P., Richter, H., Risso, C., Mahadevan, R. et al. (2011) Genome-scale dynamic modeling of the competition betweenRhodoferax and Geobacter in anoxic subsurface environments. ISME J. 5, 305–316 https://doi.org/10.1038/ismej.2010.117
44 Zomorrodi, A.R., Islam, M.M. and Maranas, C.D. (2014) d-OptCom: dynamic multi-level and multi-objective metabolic modeling of microbialcommunities. ACS Synth. Biol. 3, 247–257 https://doi.org/10.1021/sb4001307
45 Harcombe, W.R., Riehl, W.J., Dukovski, I., Granger, B.R., Betts, A., Lang, A.H. et al. (2014) Metabolic resource allocation in individual microbesdetermines ecosystem interactions and spatial dynamics. Cell Rep. 7, 1104–1115 https://doi.org/10.1016/j.celrep.2014.03.070
46 Phalak, P., Chen, J., Carlson, R.P. and Henson, M.A. (2016) Metabolic modeling of a chronic wound biofilm consortium predicts spatial partitioning ofbacterial species. BMC Syst. Biol. 10, 90 https://doi.org/10.1186/s12918-016-0334-8
47 Taffs, R., Aston, J.E., Brileya, K., Jay, Z., Klatt, C.G., McGlynn, S. et al. (2009) In silico approaches to study mass and energy flows in microbialconsortia: a syntrophic case study. BMC Syst. Biol. 3, 114 https://doi.org/10.1186/1752-0509-3-114
48 Stolyar, S., Van Dien, S., Hillesland, K.L., Pinel, N., Lie, T.J., Leigh, J.A. et al. (2007) Metabolic modeling of a mutualistic microbial community. Mol.Syst. Biol. 3, 92 https://doi.org/10.1038/msb4100131
49 Zomorrodi, A.R. and Maranas, C.D. (2012) Optcom: a multi-level optimization framework for the metabolic modeling and analysis of microbialcommunities. PLoS Comput. Biol. 8, e1002363 https://doi.org/10.1371/journal.pcbi.1002363
50 Overbeek, R., Olson, R., Pusch, G.D., Olsen, G.J., Davis, J.J., Disz, T. et al. (2014) The SEED and the Rapid Annotation of microbial genomes usingSubsystems Technology (RAST). Nucleic Acids Res. 42(Database issue), D206–D214 https://doi.org/10.1093/nar/gkt1226
51 Henry, C.S., DeJongh, M., Best, A.A., Frybarger, P.M., Linsay, B. and Stevens, R.L. (2010) High-throughput generation, optimization and analysis ofgenome-scale metabolic models. Nat. Biotechnol. 28, 977–982 https://doi.org/10.1038/nbt.1672
